Derived binomial rings I: integral Betti cohomology of log schemes

1. Problem Statement and Motivation

The paper addresses two interconnected problems in algebraic topology and arithmetic geometry:

1. Integral Homotopy Theory: While rational homotopy theory (via Sullivan's minimal models) provides a robust algebraic model for spaces, an integral version has been historically difficult to construct. Existing approaches (e.g., by Toën, Horel, Ekedahl) often rely on specific models (like cosimplicial binomial rings) or fail to capture the full structure of the singular cohomology $C^*_{\text{sing}}(X, \mathbb{Z})$ as a structured algebra over $\mathbb{Z}$. The authors seek a unified framework where the singular cochain complex of a space naturally carries a "binomial" structure that allows for the reconstruction of the space.
2. Integral Log Betti Cohomology: For a fine and saturated (fs) log complex analytic space $(X, M)$, the classical definition of log Betti cohomology involves the Kato-Nakayama space $X^{\log}$, a topological space fibered over $X$ with torus fibers. Computing the singular cohomology $R\Gamma_{\text{sing}}(X^{\log}, \mathbb{Z})$ directly is complex. Previous work (Steenbrink, Shuklin) provided formulas for $\mathbb{Q}$-coefficients using symmetric algebras, but these fail for $\mathbb{Z}$-coefficients because the free commutative algebra (or $E_\infty$-algebra) on the relevant complex is too large and does not correctly model the topology of the fibers (e.g., $S^1$).

2. Methodology

The authors introduce and utilize Derived Binomial Rings as the central algebraic tool.

• The Derived Binomial Monad ($L\text{Bin}$):

  • They define a monad $L\text{Bin}$ on the unbounded derived category $D(\mathbb{Z})$. This is a derived extension of the classical binomial monad (which acts on torsion-free abelian groups).
  • The construction proceeds via filtered monads. They first define a filtered monad $\text{Bin}_{\leq \bullet}$ on the category of lattices (free finitely generated abelian groups) using additively polynomial maps (maps $f: A \to B$ where the difference operator $D_a f$ reduces degree).
  • Using the theory of non-abelian derived functors (animation) and excisive polynomial functors (following Raksit and Brantner), they extend this to the full derived category $D(\mathbb{Z})$.
  • A Derived Binomial Ring is defined as a module over this monad $L\text{Bin}$.

• Sheafification:

  • They generalize $L\text{Bin}$ to a monad $L\text{Bin}_X$ on the $\infty$-category of $D(\mathbb{Z})$-valued sheaves on a topological space $X$. This allows for the definition of sheaves of derived binomial rings.

• The Exponential Complex:

  • For a log analytic space, they utilize the 2-term complex $\exp(\alpha) := [\mathcal{O}_X \xrightarrow{\exp} M^{\text{gr}}]$, where $M^{\text{gr}}$ is the group completion of the log structure modulo units. This complex represents the first truncation of the pushforward $R\pi_* \mathbb{Z}$ (where $\pi: X^{\log} \to X$).

3. Key Contributions and Results

A. Free Derived Binomial Rings and Eilenberg-MacLane Spaces

The authors compute the free derived binomial rings on abelian groups concentrated in a single degree.

• Coconnective Range ($n \geq 1$): For a lattice $A$ and $n \geq 1$, there is a canonical equivalence:
  $$L\text{Bin}(A^\vee[-n]) \simeq C^*_{\text{sing}}(K(A, n), \mathbb{Z})$$
  where $K(A, n)$ is the Eilenberg-MacLane space. This identifies the singular cohomology of these spaces with free derived binomial rings.
• Connective Range: In the connective range, the arithmetic of binomial rings collapses. For a torsion group $A$ and $n \geq 1$, $L\text{Bin}(A[n]) \simeq \mathbb{Z}$.
• Degree -1 for Torsion: For a finite abelian group $A$, $L\text{Bin}(A[-1])$ is 1-cotruncated. The first cohomology $L^1\text{Bin}(A[-1])$ is explicitly described (e.g., for $A=\mathbb{Z}/p$, it relates to the cyclotomic extension $K/\mathcal{O}_K$).

B. Integral Homotopy Theory

Using the above computations, the authors establish a fully faithful embedding of the category of connected nilpotent spaces of finite type into the category of derived binomial rings:
$$C^*_{\text{sing}}(-, \mathbb{Z}): \text{Spc}^{\text{op}}_{\text{nilp, ft}} \hookrightarrow \text{DBinAlg}$$

• Reconstruction: A connected nilpotent space $X$ can be reconstructed from its singular cochains as a derived binomial ring: $X \simeq \text{Map}_{\text{DBinAlg}}(C^*_{\text{sing}}(X, \mathbb{Z}), \mathbb{Z})$.
• This provides a new, efficient integral model for homotopy types, superior to classical symmetric algebras which fail integrally.

C. Integral Log Betti Cohomology Formula

The primary application is a closed formula for the integral singular cohomology of log analytic spaces.

• The Problem with Sym: The standard formula for $\mathbb{Q}$-coefficients uses the free commutative algebra: $R\pi_* \mathbb{Q} \simeq \text{Sym}^{\text{coaug}}_{\mathbb{Q}}(\exp(\alpha)_{\mathbb{Q}})$. This fails for $\mathbb{Z}$ because $\text{Sym}_{\mathbb{Z}}(\mathbb{Z}[-1])$ is too large (it has non-finitely generated cohomology).
• The Binomial Solution: The authors prove that the pushforward $R\pi_* \mathbb{Z}$ is the free coaugmented derived binomial ring on the exponential complex:
  $$R\pi_* \mathbb{Z} \simeq L\text{Bin}^{\text{coaug}}_X(\exp(\alpha))$$
  Consequently, the log Betti cohomology is given by:
  $$R\Gamma^{\log}_{\text{Betti}}(X, \mathbb{Z}) \simeq R\Gamma(X, L\text{Bin}^{\text{coaug}}_X(\exp(\alpha)))$$
  This extends Steenbrink's formula and its generalizations to integral coefficients.

D. Geometric Interpretation (Gerbes and Stacks)

The paper constructs an "algebraic approximation" $X^{\log}_H$ of the Kato-Nakayama space.

• They interpret the cosheaf of spaces $[X^{\log}, \pi]$ (assigning $U \mapsto \pi^{-1}(U)$) as a gerbe over the cosheaf of groups $(M^{\text{gr}})^\vee$.
• They define a stack $X^{\log}_H$ as a gerbe over the Cartier dual of the formal multiplicative group $\hat{\mathbb{G}}_m$ (denoted $H$).
• They show that the derived global sections of the structure sheaf of this stack recover the log Betti cohomology, providing a purely algebro-geometric model for the topological space $X^{\log}$.

4. Significance

1. Resolution of Integral Obstructions: The paper resolves the issue of modeling integral cohomology using commutative algebra. By replacing the free commutative algebra monad with the derived binomial monad, they capture the correct torsion and divisibility properties required for integral homotopy theory.
2. Unification of Topology and Arithmetic: The results link the topology of Eilenberg-MacLane spaces (and by extension, nilpotent spaces) directly to the arithmetic of binomial rings and divided powers. The deformation from the binomial ring to the divided power algebra explains the relationship between different cohomological theories.
3. Advancement in Log Geometry: The derived formula for log Betti cohomology provides a powerful computational tool that bypasses the need for the auxiliary topological space $X^{\log}$ in many contexts, expressing the cohomology directly in terms of the log scheme's structure sheaf and monoid.
4. Foundational Framework: This work (Part I) sets the stage for further developments (Part II) involving deformations, prismatic cohomology connections, and motivic lifts, positioning derived binomial rings as a fundamental structure in modern arithmetic geometry.

In summary, the paper successfully constructs a robust algebraic framework (Derived Binomial Rings) that faithfully models integral singular cohomology, enabling new computations in log geometry and providing a deeper understanding of the algebraic structure underlying topological spaces.